Subgroups — #20274386

Order 6784.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811028	C₂	1	2	yes	maximal, complemented
2	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810996	C₂	1	1	yes	maximal, characteristic
4	#16810970	C₂	1	2	yes	maximal
5	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₁₂	—	2	2	no	—
7	C₂×#11719913	—	2	2	no	—
8	#13133217	C₂²	1	1	yes	characteristic
9	#13133217	C₂²	1	2	yes	—
10	#13133214	C₂²	1	1	yes	characteristic
11	#13133151	—	2	4	no	—
12	#13133139	—	2	4	no	—
13	C₄×C₈×C₅₃	C₂²	1	2	yes	—
14	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	—	2	2	no	—
16	C₂×D₂₁₂	—	2	1	no	—
17	C₂×Dic₁₀₆	—	2	1	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	C₂×C₈×C₅₃	D₄	1	4	yes	—
20	C₂×C₈×C₅₃	D₄	1	2	yes	—
21	C₄²×C₅₃	C₂³	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	2	2	no	—
23	C₄⋊Dic₅₃	—	2	2	no	—
24	C₄⋊Dic₅₃	—	2	2	no	—
25	Dic₂₁₂	—	4	4	no	—
26	#11719913	—	4	4	no	—
27	C₂²×D₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
30	C₂×Dic₅₃	—	4	1	no	—
31	C₂×Dic₅₃	—	4	2	no	—
32	D₂₁₂	—	4	2	no	—
33	D₂₁₂	—	4	4	no	—
34	Dic₁₀₆	—	4	2	no	—
35	Dic₁₀₆	—	4	4	no	—
36	C₈×C₅₃	—	2	4	no	—
37	C₈×C₅₃	D₈	1	4	yes	—
38	C₂²×C₅₃	#127	1	1	yes	characteristic
39	C₂×D₅₃	—	8	1	no	—
40	C₂×D₅₃	—	8	2	no	—
41	C₄×C₅₃	C₂×D₈	1	2	yes	—
42	C₄×C₅₃	#137	1	4	yes	—
43	Dic₅₃	—	8	1	no	—
44	Dic₅₃	—	8	2	no	—
45	#1722	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	#493	1	1	yes	characteristic
47	C₂×C₅₃	#497	1	2	yes	—
48	D₅₃	—	16	1	no	—
49	#500	—	53	2	no	—
50	#494	—	53	1	no	—
51	#492	—	53	1	no	—
52	#486	—	53	2	no	—
53	#322	D₅₃	1	1	yes	characteristic, complemented
54	C₅₃	#1722	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₁₆	—	106	2	no	—
56	C₂×SD₁₆	—	106	2	no	—
57	C₄⋊Q₈	—	53	2	no	—
58	C₄⋊Q₈	—	53	1	no	—
59	#127	—	53	1	no	—
60	#107	—	106	4	no	—
61	#102	—	106	4	no	—
62	C₄×C₈	C₂×D₅₃	1	2	yes	—
63	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
64	C₂×Q₈	—	106	1	no	—
65	C₂×Q₈	—	106	2	no	—
66	C₂×D₄	—	106	2	no	—
67	C₂×D₄	—	106	1	no	—
68	Q₁₆	—	212	4	no	—
69	SD₁₆	—	212	4	no	—
70	C₂×C₈	D₂₁₂	1	4	yes	—
71	C₂×C₈	D₂₁₂	1	2	yes	—
72	C₄⋊C₄	—	106	2	no	—
73	C₄⋊C₄	—	106	2	no	—
74	C₄⋊C₄	—	106	2	no	—
75	C₄²	C₂²×D₅₃	1	1	yes	characteristic
76	C₂³	—	212	1	no	—
77	Q₈	—	212	2	no	—
78	Q₈	—	212	4	no	—
79	D₄	—	212	2	no	—
80	D₄	—	212	4	no	—
81	C₂×C₄	—	212	2	no	—
82	C₂×C₄	—	212	1	no	—
83	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
84	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₄₂₄	1	4	yes	—
87	C₂²	—	424	2	no	—
88	C₂²	—	424	1	no	—
89	C₂²	#13133214	1	1	yes	characteristic
90	C₄	—	424	1	no	—
91	C₄	—	424	2	no	—
92	C₄	C₂×D₄₂₄	1	2	yes	—
93	C₄	#13133250	1	4	yes	—
94	C₂	—	848	1	no	—
95	C₂	#16811000	1	1	yes	characteristic
96	C₂	#16811017	1	2	yes	—

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	#127
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1722
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16811028	C₂	semidirect
2. #16811003	C₂	semidirect
5. C₅₃×#322	C₂	semidirect
53. #322	D₅₃	semidirect
54. C₅₃	#1722	semidirect